Optimal. Leaf size=81 \[ -\frac{45 \sinh (c+d x)}{512 d (3 \cosh (c+d x)+5)}-\frac{3 \sinh (c+d x)}{32 d (3 \cosh (c+d x)+5)^2}-\frac{59 \tanh ^{-1}\left (\frac{\sinh (c+d x)}{\cosh (c+d x)+3}\right )}{1024 d}+\frac{59 x}{2048} \]
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Rubi [A] time = 0.0631422, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2664, 2754, 12, 2657} \[ -\frac{45 \sinh (c+d x)}{512 d (3 \cosh (c+d x)+5)}-\frac{3 \sinh (c+d x)}{32 d (3 \cosh (c+d x)+5)^2}-\frac{59 \tanh ^{-1}\left (\frac{\sinh (c+d x)}{\cosh (c+d x)+3}\right )}{1024 d}+\frac{59 x}{2048} \]
Antiderivative was successfully verified.
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Rule 2664
Rule 2754
Rule 12
Rule 2657
Rubi steps
\begin{align*} \int \frac{1}{(5+3 \cosh (c+d x))^3} \, dx &=-\frac{3 \sinh (c+d x)}{32 d (5+3 \cosh (c+d x))^2}-\frac{1}{32} \int \frac{-10+3 \cosh (c+d x)}{(5+3 \cosh (c+d x))^2} \, dx\\ &=-\frac{3 \sinh (c+d x)}{32 d (5+3 \cosh (c+d x))^2}-\frac{45 \sinh (c+d x)}{512 d (5+3 \cosh (c+d x))}+\frac{1}{512} \int \frac{59}{5+3 \cosh (c+d x)} \, dx\\ &=-\frac{3 \sinh (c+d x)}{32 d (5+3 \cosh (c+d x))^2}-\frac{45 \sinh (c+d x)}{512 d (5+3 \cosh (c+d x))}+\frac{59}{512} \int \frac{1}{5+3 \cosh (c+d x)} \, dx\\ &=\frac{59 x}{2048}-\frac{59 \tanh ^{-1}\left (\frac{\sinh (c+d x)}{3+\cosh (c+d x)}\right )}{1024 d}-\frac{3 \sinh (c+d x)}{32 d (5+3 \cosh (c+d x))^2}-\frac{45 \sinh (c+d x)}{512 d (5+3 \cosh (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.187694, size = 58, normalized size = 0.72 \[ \frac{59 \tanh ^{-1}\left (\frac{1}{2} \tanh \left (\frac{1}{2} (c+d x)\right )\right )-\frac{3 (182 \sinh (c+d x)+45 \sinh (2 (c+d x)))}{(3 \cosh (c+d x)+5)^2}}{1024 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 108, normalized size = 1.3 \begin{align*} -{\frac{9}{512\,d} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +2 \right ) ^{-2}}+{\frac{69}{1024\,d} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +2 \right ) ^{-1}}+{\frac{59}{2048\,d}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +2 \right ) }+{\frac{9}{512\,d} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -2 \right ) ^{-2}}+{\frac{69}{1024\,d} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -2 \right ) ^{-1}}-{\frac{59}{2048\,d}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -2 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03263, size = 169, normalized size = 2.09 \begin{align*} -\frac{59 \, \log \left (3 \, e^{\left (-d x - c\right )} + 1\right )}{2048 \, d} + \frac{59 \, \log \left (e^{\left (-d x - c\right )} + 3\right )}{2048 \, d} - \frac{3 \,{\left (241 \, e^{\left (-d x - c\right )} + 295 \, e^{\left (-2 \, d x - 2 \, c\right )} + 59 \, e^{\left (-3 \, d x - 3 \, c\right )} + 45\right )}}{256 \, d{\left (60 \, e^{\left (-d x - c\right )} + 118 \, e^{\left (-2 \, d x - 2 \, c\right )} + 60 \, e^{\left (-3 \, d x - 3 \, c\right )} + 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + 9\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.17252, size = 1661, normalized size = 20.51 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.73531, size = 445, normalized size = 5.49 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15086, size = 123, normalized size = 1.52 \begin{align*} \frac{59 \, \log \left (3 \, e^{\left (d x + c\right )} + 1\right )}{2048 \, d} - \frac{59 \, \log \left (e^{\left (d x + c\right )} + 3\right )}{2048 \, d} + \frac{3 \,{\left (59 \, e^{\left (3 \, d x + 3 \, c\right )} + 295 \, e^{\left (2 \, d x + 2 \, c\right )} + 241 \, e^{\left (d x + c\right )} + 45\right )}}{256 \, d{\left (3 \, e^{\left (2 \, d x + 2 \, c\right )} + 10 \, e^{\left (d x + c\right )} + 3\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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